Properties of...

Properties of Orthogonality of two circles

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Q. Two given circles

x^{2} + y^{2} + a x + b y + c = 0

and

x^{2} + y^{2} + d x + e y + f = 0

will intersect each other orthogonally, only when

2ad+2be = c+f
a+b+c = d+e+f
ad+be = c+f

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Find the equation of the circle orthogonal to the circles $x^{2} + y^{2} + 3 x - 5 y + 6 = 0 a n d 4 x^{2} + 4 y 2 - 28 x + 29 = 0$ and whose center lies on the line 3x + 4y + 1 = 0.

2x² + 2y² + y - 29 = 0
x² + y² + y/4 - 29/4 = 0
8x² + 8y² + 2y - 29 = 0
4x² + 4y² + 2y - 29 = 0

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Two circles $x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y + c_{1} = 0 a n d x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y + c^{2} = 0$ are said to be orthogonal. Then $2 g_{1} g_{2} + 2 f_{1} f_{2} = c_{1} + c_{2}$

True
False

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Q. A circle S passes through the point (0, 1) and is orthogonal to the circles

(x - 1)^{2} + y^{2} = 16

and

x^{2} + y^{2} = 1

then,

Radius of S is 8
Radius of S is 7
Centre of S is (-7, 1)
Center of S is (-8, 1)

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